Skip to main content
SpringerLink
Log in

Relativized Adjacency

  • Published:
Journal of Logic, Language and Information Aims and scope Submit manuscript

Abstract

For each class in the piecewise-local subregular hierarchy, a relativized (tier-based) variant is defined. Algebraic as well as automata-, language-, and model-theoretic characterizations are provided for each of these relativized classes, except in cases where this is provably impossible. These various characterizations are necessarily intertwined due to the well-studied logic-automaton connection and the relationship between finite-state automata and (syntactic) semigroups. Closure properties of each class are demonstrated by using automata-theoretic methods to provide constructive proofs for the closures that do hold and giving language-theoretic counterexamples for those that do not. The net result of all of this is that, rather than merely existing as an operationally-defined parallel set of classes, these relativized variants integrate cleanly with the other members of the piecewise-local subregular hierarchy from every perspective. Relativization may even prove useful in the characterization of star-free, as every star-free stringset is the preprojection of another (also star-free) stringset whose syntactic semigroup is not a monoid.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Notes

  1. Because our notion of structural containment is distinct from the standard model-theoretic notion of substructures, we are careful to avoid that term.

  2. The values of Q and \(\varSigma \) are implied by the signature of \(\delta \). Some prefer to specify them explicitly, which would result in automata being 5-tuples.

  3. Since the result is not necessarily canonical, a larger (thus a smaller \(T\)) may also exist.

  4. Software available at https://github.com/vvulpes0/Language-Toolkit-2.

References

  • Aho, A. V., Garey, M. R., & Ullman, J. D. (1972). The transitive reduction of a directed graph. SIAM Journal on Computing, 1(2), 131–137. https://doi.org/10.1137/0201008

    Article  Google Scholar 

  • Aksënova, A., Deshmukh, S. (2018). Formal restrictions on multiple tiers. In Proceedings of the Society for Computation in Linguistics, Salt Lake City, Utah, 1, 64–73. https://doi.org/10.7275/R5K64G8S

  • Beauquier, D., Pin, J.-E. (1989). Factors of words. In G. Ausiello, M. Dezani-Ciancaglini, S. Ronchi Della Rocca (Eds.) Automata, languages and programming: 16th International Colloquium, Lecture Notes in Computer Science, vol 372, Springer Berlin/Heidelberg (pp. 63–79), https://doi.org/10.1007/BFb0035752

  • Beesley, K. R., & Karttunen, L. (2003). Finite state morphology. CSLI Publications.

    Google Scholar 

  • Brzozowski, J. A., & Simon, I. (1973). Characterizations of locally testable events. Discrete Mathematics, 4(3), 243–271. https://doi.org/10.1016/S0012-365X(73)80005-6

    Article  Google Scholar 

  • Caron, P. (1998). LANGAGE: A maple package for automaton characterization of regular languages. In D. Wood, S. Yu (Eds.) Automata implementation, lecture notes in computer science, vol 1436, Springer Berlin/Heidelberg (pp. 46–55), https://doi.org/10.1007/BFb0031380

  • Chandlee, J. (2014). Strictly local phonological processes. PhD thesis, University of Delaware, https://chandlee.sites.haverford.edu/wp-content/uploads/2015/05/Chandlee_dissertation_2014.pdf

  • Cser, A. (2010). The -alis/-aris allomorphy revisited. In F. Rainer, W. Dressler, D. Kastovsky, H. C. Luschützky (Eds.) Variation and Change in Morphology: Selected Papers from the 13th International Morphology Meeting, John Benjamins Publishing Company, Vienna, Austria (pp. 33–52), https://doi.org/10.1075/cilt.310.02cse

  • De Luca, A., & Restivo, A. (1980). A characterization of strictly locally testable languages and its application to subsemigroups of a free semigroup. Information and Control, 44(3), 300–319. https://doi.org/10.1016/S0019-9958(80)90180-1

    Article  Google Scholar 

  • De Santo, A., Graf, T. (2019). Structure sensitive tier projection: Applications and formal properties. In R. Bernardi, G. Kobele, S. Pogodalla (Eds.) Formal Grammar 2019, lecture notes in computer science, vol 11668, Springer Verlag (pp. 35–50), https://doi.org/10.1007/978-3-662-59648-7_3

  • Edlefsen, M., Leeman, D., Myers, N., Smith, N., Visscher, M., Wellcome, D. (2008). Deciding strictly local (SL) languages. In J. Breitenbucher (Ed.) Proceedings of the 2008 Midstates Conference for Undergraduate Research in Computer Science and Mathematics, (pp. 66–73).

  • Gold, E. M. (1967). Language identification in the limit. Information and Control, 10(5), 447–474. https://doi.org/10.1016/S0019-9958(67)91165-5

    Article  Google Scholar 

  • Graf, T. (2017). The power of locality domains in phonology. Phonology, 34(2), 385–405. https://doi.org/10.1017/S0952675717000197

    Article  Google Scholar 

  • Haines, L. H. (1969). On free monoids partially ordered by embedding. Journal of Combinatorial Theory, 6(1), 94–98. https://doi.org/10.1016/s0021-9800(69)80111-0

    Article  Google Scholar 

  • Heinz, J., Rawal, C., Tanner, H. G. (2011). Tier-based strictly local constraints for phonology. In Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Short Papers, Association for Computational Linguistics, Portland, Oregon, 2, 58–64, https://aclweb.org/anthology/P11-2011

  • Hulden, M. (2009). Finite-state machine construction methods and algorithms for phonology and morphology. PhD thesis, The University of Arizona.

  • Jardine, A., & Heinz, J. (2016). Learning tier-based strictly 2-local languages. Transactions of the Association for Computation in Linguistics, 4, 87–98. https://doi.org/10.1162/tacl_a_00085

    Article  Google Scholar 

  • Jardine, A., McMullin, K. (2017). Efficient learning of tier-based strictly k-local languages. In F. Drewes, C. Martín-Vide, B. Truthe (Eds.) Language and automata theory and applications: 11th international conference, lecture notes in computer science, vol 10168, Springer, Cham (pp. 64–76), https://doi.org/10.1007/978-3-319-53733-7_4

  • Ji, J., Heinz, J. (2020). Input strictly local tree transducers. In A. Leporati, C. Martín-Vide, D. Shapira, C. Zandron (Eds.) Language and automata theory and applications. In Proceedings of the 14th international conference, LATA 2020, Springer International Publishing, Cham, Switzerland, Theoretical Computer Science and General Issues, vol 12038, (pp. 369–381), https://doi.org/10.1007/978-3-030-40608-0_26

  • Knuth, D. E., Morris, J. H., & Pratt, V. R. (1977). Fast pattern matching in strings. SIAM Journal on Computing, 6(2), 323–350. https://doi.org/10.1137/0206024

    Article  Google Scholar 

  • Lambert, D., Rogers, J. (2020). Tier-based strictly local stringsets: Perspectives from model and automata theory. In Proceedings of the society for computation in linguistics, New Orleans, Louisiana, 3, 330–337, https://doi.org/10.7275/2n1j-pj39

  • Libkin, L. (2004). Elements of finite model theory. Texts in Theoretical Computer Science. https://doi.org/10.1007/978-3-662-07003-1

    Article  Google Scholar 

  • McMullin, K. (2016). Tier-based locality in long-distance phonotactics: Learnability and typology. PhD thesis, University of British Columbia.

  • McNaughton, R. (1974). Algebraic decision procedures for local testability. Mathematical Systems Theory, 8(1), 60–76. https://doi.org/10.1007/bf01761708

    Article  Google Scholar 

  • McNaughton, R., & Papert, S. A. (1971). Counter-free automata. MIT Press.

    Google Scholar 

  • Nerode, A. (1958). Linear automaton transformations. In Proceedings of the American Mathematical Society, American Mathematical Society, 9, 541–544, https://doi.org/10.1090/s0002-9939-1958-0135681-9

  • Pin, J.-E. (1997). Syntactic semigroups. In G. Rozenberg, A. Salomaa (Eds.) Handbook of Formal Languages: Volume 1 Word, Language, Grammar, Springer-Verlag, Berlin (pp. 679–746), https://doi.org/10.1007/978-3-642-59136-5_10

  • Rogers, J. (1996). A model-theoretic framework for theories of syntax. In Proceedings of the 34th Annual Meeting of the Association for Computational Linguistics, Association for Computational Linguistics, Santa Cruz, CA, (pp. 10–16), https://doi.org/10.3115/981863.981865

  • Rogers, J. (1997). Strict LT\({}_2\) : regular : local : recognizable. In C. Retoré (Ed.) Logical aspects of computational linguistics: first international conference, LACL ’96 (Selected Papers), Springer-Verlag, Berlin, Lecture Notes in Computer Science, vol 1328, (pp. 366–385), https://doi.org/10.1007/BFb0052167

  • Rogers, J. (1998). A descriptive approach to language-theoretic complexity. (Monograph.) Studies in Logic, Language, and Information, CSLI Publications.

  • Rogers, J., & Lambert, D. (2019). Extracting Subregular constraints from Regular stringsets. Journal of Language Modelling, 7(2), 143–176. https://doi.org/10.15398/jlm.v7i2.209

    Article  Google Scholar 

  • Rogers, J., Lambert, D. (2019b). Some classes of sets of structures definable without quantifiers. In Proceedings of the 16th Meeting on the Mathematics of Language, Association for Computational Linguistics, Toronto, Canada, (pp. 63–77), https://doi.org/10.18653/v1/W19-5706

  • Rogers, J., & Pullum, G. K. (2011). Aural pattern recognition experiments and the subregular hierarchy. Journal of Logic, Language and Information, 20(3), 329–342. https://doi.org/10.1007/s10849-011-9140-2

    Article  Google Scholar 

  • Rogers, J., Heinz, J., Bailey, G., Edlefsen, M., Visscher, M., Wellcome, D., Wibel, S. (2010). On languages piecewise testable in the strict sense. In C. Ebert, G. Jäger, J. Michaelis (Eds.) The mathematics of language: Revised selected papers from the 10th and 11th biennial conference on the mathematics of language, LNCS/LNAI, vol 6149, FoLLI/Springer (pp. 255–265), https://doi.org/10.1007/978-3-642-14322-9_19

  • Schützenberger, M.-P. (1965). On finite monoids having only trivial subgroups. Information and Control, 8(2), 190–194. https://doi.org/10.1016/s0019-9958(65)90108-7

    Article  Google Scholar 

  • Schützenberger, M.-P. (1975). Sur certaines opérations de fermeture dans les langages rationnels. Symposia Mathematica, 15, 245–253.

    Google Scholar 

  • Simon, I. (1975). Piecewise testable events. In H. Brakhage (Ed.) Automata theory and formal languages, lecture notes in computer science, vol 33, Springer-Verlag, Berlin (pp. 214–222), https://doi.org/10.1007/3-540-07407-4_23

Download references

Author information

Authors and Affiliations

  1. Université Jean Monnet Saint-Etienne, CNRS, Institut d Optique Graduate School, Laboratoire Hubert Curien UMR 5516, 42023, Saint-Etienne, France

    Dakotah Lambert

Authors
  1. Dakotah Lambert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dakotah Lambert.

Ethics declarations

Conflict of interest

The author has no conflicts of interest to declare that are relevant to the content of this article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lambert, D. Relativized Adjacency. J of Log Lang and Inf 32, 707–731 (2023). https://doi.org/10.1007/s10849-023-09398-x

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10849-023-09398-x

Keywords

Mathematics Subject Classification

Search

Navigation